Question

Water from a pipe is coming at a rate of 100 litres per minute. If the radius of the pipe is 5 cm, the Reynolds number for the flow is of the order of: (density of water = 1000 kg/m³, coefficient of viscosity of water = 1 mPa s)
1. 10³
2. 10⁴
3. 10²
4. 10⁶

Answer 1

Correct Answer - Option 2 : 10⁴

Concept:

In Fluid Mechanics, Reynolds number is used to represent the flow of fluid. It is dimensionless. It is defined as the ratio of inertial force to the viscous force.

Calculation:

We know that, Reynolds number,

\(Re=\frac{\rho \text{vd}}{\eta }\)     ----(1)

Where, v = Velocity of the fluid with respect to the object

d = Diameter of the pipe

ρ = Density of the fluid

η = Viscosity of the fluid

In the question, the velocity is given as volume velocity in litres per minute (L/m). Its unit is m³/s.

Since, we need m/s (velocity), we need to find the velocity using below given formula:

Volume velocity = velocity × area = v × πr²

\(\Rightarrow v=\frac{Volume~velocity}{\pi {{r}^{2}}}\) 

\(\Rightarrow v=\frac{100\times {{10}^{-3}}}{60}\times \frac{1}{\pi \times {{\left( 5\times {{10}^{-2}} \right)}^{2}}}\) 

∴ v = 0.21 m/s

On substituting the given values in equation (1), we get,

\(\text{Reynolds }\!\!~\!\!\text{ number}=\frac{1000\times 0.21\times \left( 2\times 5\times {{10}^{-2}} \right)}{1\times {{10}^{-3}}}\) 

Note: 1 mPa s = 1 milli-pascal second

Reynolds number = 21000 = 2.1 × 10⁴

The order of the Reynolds number is 10⁴.